Fitting GP to lightcurve 428_…in Stan
Notebook outlining the fitting of GP to thunderKAT lightcurve ID$ 428_…
- The light curve has \(N =\) 21 observations over a range of 147.5681816 days.
- The mean flux density is \(\bar{y} =\) 0.1903983 Jy.
- The mean standard error is 0.1903983 Jy.
- The observational noise is faint relative to the brightness of the observations.
- Observations are evenly spread over the time range.
- There are hints of correlated noise components in the light curve.
Basic Model
- Zero constant mean function
- Homoskedastic noise
- Not using data on error in observations
- Wide prior on observational errors
\[y \sim \mathcal{N}(f(x), \sigma_\textrm{noise}^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell^2}\right\}\]
\[\ell \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_\textrm{noise} \sim \mathcal{N}^+(0,1)\]
MCMC Results
variable mean median sd mad q5 q95
eta 0.184588 0.152648 0.113476 0.065523 0.083235 0.394774
ell 150.580662 132.664000 78.972809 60.348195 63.226605 295.029000
sigma 0.006285 0.006116 0.001137 0.001059 0.004713 0.008371
rhat ess_bulk ess_tail
0.999966 2961 2328
1.001321 2922 2491
1.000792 3212 2674
Posterior Predictive Samples
The fitted model has a very long lengthscale, comparable to the length of the observational window. The estimated observational noise has a standard deviation more than an order of magnitude of that in the original data. The combination of these parameters has lead to a very smooth fit that passes through the middle of the observed data points rather than through any datapoints themselves.
Observational Errors Model
- Zero constant mean function
- Include data on error in observations of \(y_i\)
- Heteroskedastic (Gaussian) noise
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell^2}\right\}\]
\[\ell \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
MCMC Results
variable mean median sd mad q5 q95 rhat
eta 0.139203 0.135184 0.027459 0.025415 0.102757 0.189866 0.999570
ell 11.188619 11.204300 0.493513 0.472801 10.334480 11.964810 1.001436
sigma[1] 0.000393 0.000393 0.000042 0.000042 0.000326 0.000461 1.004134
ess_bulk ess_tail
6179 3134
5518 3015
8251 2818
Posterior Predictive Samples
By including the observed observational errors for setting priors on the Gaussian noise of each observation, the fitted median passes through each of the observed points.
Constant (non-zero) Mean function
- Constant mean function
- Prior on mean function value
- Include data on error in observations of \(y_i\)
- Heteroskedastic (Gaussian) noise
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(C, k(x, x'))\]
\[C \sim \mathcal{U}[0,1]\]
\[k(x,x') = \eta^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell^2}\right\}\]
\[\ell \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
MCMC Results
variable mean median sd mad q5 q95 rhat
eta 0.007869 0.007660 0.001386 0.001252 0.005995 0.010521 1.000645
ell 1.562706 1.141115 1.230891 0.583181 0.557737 4.617188 1.001045
C 0.190356 0.190383 0.001761 0.001681 0.187439 0.193192 1.001547
sigma[1] 0.000393 0.000392 0.000041 0.000041 0.000324 0.000461 1.000504
ess_bulk ess_tail
3322 1919
2440 1297
3393 1961
4224 2856
Posterior Predictive Samples
Fixed constant (non-zero) Mean function
- Constant mean function set at fixed value, e.g., mean of observations
- Include data on error in observations of \(y_i\)
- Heteroskedastic (Gaussian) noise
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(C, k(x, x'))\]
\[k(x,x') = \eta^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell^2}\right\}\]
\[\ell \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
Mean Function = 0.2
Mean Function = 0.195
Mean Function = 0.18
Matern 3/2 kernel, zero mean
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta^2 \left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell}\right\}\]
\[\ell \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
MCMC Results
variable mean median sd mad q5 q95 rhat
eta 0.154295 0.138354 0.064650 0.043151 0.090575 0.273144 1.000380
ell 77.002830 72.413850 22.854942 18.158514 49.429170 120.076800 1.000103
sigma[1] 0.000393 0.000393 0.000041 0.000041 0.000327 0.000460 1.000717
ess_bulk ess_tail
4108 2402
3854 2180
7848 2658
Posterior Predictive Samples
SE + Matern 3/2 additive kernel
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta^2 \left[ \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\} + \left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right\} \right]\]
\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]
\[\ell_\mathrm{M} \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
variable mean median sd mad q5 q95 rhat
eta 0.010996 0.008687 0.008222 0.004604 0.003972 0.025317 1.000166
ell_SE 40.137106 34.952250 21.529021 13.219603 19.576910 75.923105 1.000316
ell_M 54.647213 52.715750 14.041654 12.820265 35.660370 80.404210 1.000264
sigma[1] 0.000394 0.000394 0.000041 0.000039 0.000327 0.000461 0.999895
ess_bulk ess_tail
3901 2446
4500 2262
3855 2547
6845 2694Posterior Predictive Samples
SE x Matern 3/2 multiplicative kernel
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\}\left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right\}\]
\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]
\[\ell_\mathrm{M} \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
MCMC Results
variable mean median sd mad q5 q95 rhat
eta 0.037945 0.035572 0.023139 0.017537 0.011372 0.072392 1.170342
ell_SE 21.972784 1.705505 29.944829 1.558493 0.592920 77.967890 1.619411
ell_M 30.591427 1.660200 38.765222 1.471168 0.601296 97.911300 1.635714
sigma[1] 0.000393 0.000393 0.000041 0.000042 0.000323 0.000461 1.001599
f_star[1] 0.183928 0.183926 0.000398 0.000388 0.183284 0.184594 1.001590
ess_bulk ess_tail
16 65
6 78
6 71
5349 2490
3628 3852
Posterior Predictive Samples
Matern 3/2 + QP kernel
\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]
\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]
\[k(x,x') = \eta^2 \left[ \exp\left\{ -\frac{2 \sin^2\left( \pi\frac{\sqrt{(x - x')^2}}{T}\right)}{\ell_\mathrm{P}^2}\right\} + \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\} + \left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right\} \right]\]
\[\ell_\mathrm{P} \sim \mathrm{InvGamma}(5,5)\]
\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]
\[\ell_\mathrm{M} \sim \mathrm{InvGamma}(5,5)\]
\[\eta \sim \mathcal{N}^+(0,1)\]
\[T \sim U(\textrm{min_gap}(\boldsymbol{x}), \textrm{range}(\boldsymbol{x}))\]
\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]
variable mean median sd mad q5 q95 rhat ess_bulk
eta 0.0040 0.0031 0.0035 0.0015 0.0015 0.0083 1.0707 37
ell_SE 23.6680 22.3852 15.9193 10.2359 0.9128 47.8225 1.2686 10
ell_M 32.3308 34.6120 16.2567 10.8554 0.9071 54.1578 1.2838 10
ell_P 2.5511 2.2592 1.2934 0.8951 1.2297 5.0154 1.0145 394
T 103.3865 110.6080 34.2538 33.0143 27.5613 144.0561 1.0606 44
ess_tail
244
15
17
2197
46
Posterior Predictive Samples
